The battery powers a small memory that keeps track of settings you may have made in the BIOS configuration (During a boot, hit DEL or F1 or whatever when prompted to get into the configuration menu). The settings are a mixture of trivial stuff, such as whether you want the Num Lock key on by default, and critical stuff, such as boot sequence, memory timings, etc. One other use of that memory, as you found out, is to store the current time across boots. If that memory loses power (battery dead, no external power), then the BIOS uses factory defaults and tries to sense the environment as best it can.
In the old days, if you lost the BIOS battery, the computer typically would not boot. That was because things like the disk drive characteristics were stored in the BIOS, and the BIOS needs that information to access the hard drive. To boot, you needed to dig out the manual(s) and input the proper settings by hand (much harder than fixing date/time). These days, the BIOS is able to talk to the disk drive controller and figure out those settings on its own.
I have a teenager around here somewhere, will bring the teen in to read that and splain it all to me! :)
Seriously, although it is still a tad above my level, it is the most nearly understandable thing I have yet read on the subject — and I spent a LOT of time trying to grasp this. Many thanks again, cynwoody!